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Properties

Label 2-21e2-63.5-c1-0-20
Degree $2$
Conductor $441$
Sign $0.790 + 0.612i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.424i·2-s + (0.625 − 1.61i)3-s + 1.82·4-s + (1.80 + 3.12i)5-s + (−0.685 − 0.265i)6-s − 1.62i·8-s + (−2.21 − 2.01i)9-s + (1.32 − 0.765i)10-s + (−3.20 − 1.85i)11-s + (1.13 − 2.94i)12-s + (5.23 + 3.02i)13-s + (6.17 − 0.960i)15-s + 2.95·16-s + (0.532 + 0.921i)17-s + (−0.856 + 0.941i)18-s + (−3.16 − 1.82i)19-s + ⋯
L(s)  = 1  − 0.299i·2-s + (0.360 − 0.932i)3-s + 0.910·4-s + (0.806 + 1.39i)5-s + (−0.279 − 0.108i)6-s − 0.572i·8-s + (−0.739 − 0.673i)9-s + (0.419 − 0.241i)10-s + (−0.967 − 0.558i)11-s + (0.328 − 0.848i)12-s + (1.45 + 0.838i)13-s + (1.59 − 0.248i)15-s + 0.738·16-s + (0.129 + 0.223i)17-s + (−0.201 + 0.221i)18-s + (−0.725 − 0.418i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.790 + 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98081 - 0.678130i\)
\(L(\frac12)\) \(\approx\) \(1.98081 - 0.678130i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.625 + 1.61i)T \)
7 \( 1 \)
good2 \( 1 + 0.424iT - 2T^{2} \)
5 \( 1 + (-1.80 - 3.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.20 + 1.85i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.23 - 3.02i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.532 - 0.921i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.16 + 1.82i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.314 - 0.181i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.857 - 0.495i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.08iT - 31T^{2} \)
37 \( 1 + (-4.00 + 6.93i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.09 - 3.62i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.89 + 3.28i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.67T + 47T^{2} \)
53 \( 1 + (3.92 - 2.26i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 0.0275iT - 61T^{2} \)
67 \( 1 + 9.72T + 67T^{2} \)
71 \( 1 - 5.55iT - 71T^{2} \)
73 \( 1 + (1.95 - 1.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 6.53T + 79T^{2} \)
83 \( 1 + (-1.52 - 2.64i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.47 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.67 + 0.964i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.97776333821919005227615419159, −10.51876946678355892768002880850, −9.260259893741086501814784452600, −8.103031987041336834592023307336, −7.17065438401302999120378656192, −6.31945464329444014217380571343, −5.96470765963367201191753680258, −3.49919518549673075393299219813, −2.63937253275272363957479338320, −1.72579783184673203744013813884, 1.74708540445170543951946496090, 3.11375409439657999116113069726, 4.64112024613672688387710953224, 5.48378711470846055655311594799, 6.21647267257772224593285034515, 7.959832512244129326475534662860, 8.368298396208431922756333547203, 9.430592133154988221729290434667, 10.31788928065807599055409018783, 10.91807039083219004109512095714

Graph of the $Z$-function along the critical line